The fractional differencing enables the degree of differencing to take any real value rather than being restricted to integer values. The fractionally differenced processes are capable of modeling long-term persistence. The process

is known as a fractional Gaussian noise process or an ARFIMA() process, where , is a white noise process with mean zero and variance , and is the backshift operator such that . The extension of an ARFIMA() model combines fractional differencing with an ARMA() model, known as an ARFIMA() model.

Consider an ARFIMA() represented as where . With the following statements you can

• generate the simulated 300 observations data

• obtain the fractionally differenced data

• compute the autocovariance function

• compute the log-likelihood function

• fit a fractionally integrated time series model to the data

proc iml;
/* ARFIMA(0,0.4,0) */
lag = (0:12)`;
call farmacov(autocov_D_IS_04, 0.4); 
call farmacov(D_IS_005, 0.05); 
print lag autocov_D_IS_04 D_IS_005;  
d = 0.4;
call farmasim(yt, d) n=300 sigma=2 seed=5345; 
call fdif(zt, yt, d); 
*print zt;
call farmalik(lnl, yt, d); 
print lnl;
call farmafit(d, ar, ma, sigma, yt); 
print d sigma;

Output 13.4.9 Plot of Generated ARFIMA(0,0.4,0) Process (FARMASIM)

The FARMASIM function generates the data shown in Figure 13.4.9.

Output 13.4.10 Plot of Fractionally Differenced Process (FDIF)

The FDIF function creates the fractionally differenced process. Figure 13.4.10 shows a white noise series.

The first column is the autocovariance function of the ARFIMA(0,0.4,0) model, and the second column is the autocovariance function of the ARFIMA(0,0.05,0) model. The first column decays to zero more slowly than the second column.

The first row value is the log-likelihood function of the ARFIMA(0,0.4,0) model. Because the default option of the estimates method is the conditional sum of squares, the last two rows of Figure 13.4.12 contain missing values.

The final estimates of the parameters are and , while the true values of the data generating process are and .